Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on λ-th power skewness and kurtosis

Abstract

Temperature data, like many other measurements in quantitative fields, are usually modelled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution EPD_(λ)(µ, σ) including tests for normality and Laplacity when λ is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two Z-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain Z-scores that are nearly independent and distributed as standard Gaussians, resulting in a X^(2)_(2) distribution valid for any sample size (up to very high precision for n ≥ 20). The case λ = 1 leads to a powerful test of fit for the Laplace(µ, σ) distribution, whose empirical power is superior to all 39 competitors in the literature, over a wide range of 400 alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.